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Adv Alegbra HW Solutions 62 - Q(ii Prove that − I is the...

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62 It follows that | G |= 2 n ,for A has order n . (iv) Prove that each matrix in G has a unique expression of the form B i A j , where i = 0 , 1 and 0 j < n . Conclude that | G |= 2 n and that G = D 2 n . Solution. Note f rst that B / ∈h A i , for every matrix in h A i is diag- onal and B is not diagonal. Suppose that B i A j = B k A ` , where the exponents lie in the proper ranges. If i = j , then we may can- cel to obtain A j = A ` . Since A has order n ,wehave n | j ` . But | j ` | < n , and so j = ` .I f i ±= j , then b ∈h A i , and we have already observed that this is not so. It follows that G contains exactly 2 n elements. 2.86 Recall that the group of quaternions Q (de f ned in Example 2.98) consists of the 8 matrices in GL ( 2 , C ) , Q ={ I , A , A 2 , A 3 , B , BA , BA 2 , BA 3 } , where A = £ 01 10 ± and B = £ 0 i i 0 ± . (i) Prove that Q is a nonabelian group with operation matrix multipli- cation. Solution. We merely organize the needed calculations. First, show that Q is closed under multiplication in the same way as D 2 n was shown to be closed in the previous exercise. De f ne X ={ A i : 0 i < 4 } and Y ={ BA i : 0 i < 4 } , and show that XX Q , XY Q , etc. Second, the inverse of each matrix M Q
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Unformatted text preview: Q . (ii) Prove that − I is the only element in Q of order 2, and that all other elements M ±= I satisfy M 2 = − I . Solution. Straightforward multiplication. (iii) Show that Q has a unique subgroup of order 2, and it is the center of Q . Solution. A group of order 2 must be a cyclic group generated by an element of order 2. It is shown, in part (i), that − I is the only element of order 2. It is clear that h− I i ≤ Z ( Q ) , for scalar matrices commute with every matrix. On the other hand, for every element M ±= ² I , there is N ∈ Q with M N ±= N M . (iv) Prove that h− I i is the center Z ( Q ) . Solution. For each M ∈ Q but not in h− I i , there is a matrix M ∈ Q with M M ±= M M . 2.87 Prove that the quaternions Q and the dihedral group D 8 are nonisomorphic groups of order 8....
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